Modeling Cardiac Electric Fields

Olaf Dössel and Gunnar Seemann

Institute of Biomedical Engineering, Universität Karlsruhe (TH), Karlsruhe, Germany

Correspondence: Olaf Dössel, Institute of Biomedical Engineering, Universität Karlsruhe (TH), Kaiserstr. 12, 76128 Karlsruhe, Germany.
E-mail: olaf.doessel@ibt.uni-karlsruhe.de, phone +49 721 608 2650, fax +49 721 608 2789

1.  Introduction

Electrophysiological modeling of the human heart – that means to understand all processes of electric depolarization and repolarization in the heart:

-  starting in the Sinus Node and ending with repolarization in the ventricles,
-  starting with all individual ion channels end ending with the Electrocardiogram (ECG),
-  starting with the physiological case ("healthy") and ending with various pathologies ("diseases"),
-  starting with an isolated heart and ending with the heart as a part of a complex control circuit.

The motivation is to understand what's going on ("basic research"). But on the long run important applications arise. Computer models can become a tool for:

- education of cardiologist. Complex interdependencies of parameters in the heart can be explained and visualized.
- training of cardiologist. Annotation or even RF-ablation can be carried out in virtual hearts until the physician is perfect.
- detailed diagnostics. Hypotheses about the basic origin of a disease of a patient can be "plugged" into the virtual heart of an individual patient and the resulting intracardial or extracorporal signals can be compared with reality.
- therapy planning. Pharmacological therapies, RF-ablation, pace maker settings or heart surgery can be tested with the virtual heart of the individual patient until the optimum is reached.
- finding new types of therapy. Better understanding can lead to substantial changes of paradigms.

This brief overview of a very dynamic field of research will give a short description about the fundamental mathematical models and then concentrate on new results concerning cell models, coupling of cells, external stimulation and validation.

2.  Mathematical Modeling of the Electrophysiology of the Heart

Mathematical modeling of the electrophysiological properties of cells is based on the Hodgkin-Huxley-approach [ Hodgkin and Huxley, 1952 ]. Next step to more specific models of cardiomyocytes was the Beeler Reuter model, with included already 4 ion channels [ Beeler and Reuter, 1977 ]. FitzHugh and Nagumo [ FitzHugh, 1961 ] proposed a simplified model that leads to reasonable results of the transmembrane voltage V_m with low calculation times. Luo and Rudy [ Luo and Rudy, 1994 ] did pioneering work in further detailed description of ventricular cells. Today about 20 cell models are published [see a list e.g. in Dössel et al., 2002 ], that include e.g. specific behavior of the diadic space and intracellular structures like the sarcoplasmatic reticulum. Also metabolism and deformation are considered in detail in some models. The basic mathematical structure is:

                                        (1)

where I_m is the transmembrane current, C_m the membrane capacitance, V_m the transmembrane voltage, I_{ion i} the current through ion channel i, and I_stim the stimulus current. Every current through a specific type of ion channel is described as:

                                                                                                  (2)

where V_{Nernst i} is the Nernst-Voltage of ion i and G_{ion i} the conductance of ion channel type i. Now finally every conductance of an ion channel is described by a nonlinear differential equation. For sodium e.g. the equation looks like:

                                      (3)

where m is the fraction of m-particles in the open state, h the fraction of h-particles in the open state, a_m, b_m, a_h, b_h rate constants and G_{Na max} the maximum conductance.

Via equation (1) all ion channels in one cell are coupled. The set of sometimes up to 40 coupled differential equations is solved using e.g. a one step Euler approximation. Typical time steps are in the order of 1µs to 100µs. Perfect simulation of all cellular properties requires precise knowledge of all rate constants and maximum conductance values. All these values may be different depending on where the tissue is located in the heart and whether the heart is healthy or not.

The coupling of millions of cells to build a patch of myocardium or even the complete heart is done using e.g. the bidomain model [ Geselowitz, 1983; Plonsey and Barr, 1987 ]. Coupled Poisson equations for the extracellular potential F_e and intracellular potential F_i can be transformed into two differential equations for F_e and V_m [see e.g. Henriquez, 1993; Hooke et al., 1994 ]:

                                                                                        (4)

                                                                                            (5)

Where g_i and g_e are bidomain intra- and extracellular conductivity tensors and b the cell surface to cell volume relation. The numerical solution is usually done by first solving the Poisson equation (4) to deliver F_e while assuming the V_m is known throughout the volume. Then equation (5) is solved delivering V_m for the next time step assuming F_e is known.

Possible simplifications are monodomain approaches that go without the time consuming poisson equation. They are good in case of equal anisotropy ratios, but this assumption is not accepted in general any more. Another simplification is the neglection of local fiber direction or the neglection of the twist of fiber direction in the myocardium. Also 2D-models are presented e.g. for the atria.

Perfect simulation of the spread of depolarization requires precise knowledge of intracellular and extracellular bidomain conductivity tensors, fiber direction and geometry of the heart. Reasonable boundary conditions have to be included e.g. for the boundary heart to blood.

3.  Cell Models

New refinements of cell models focus on a better

-  translation of data of various mammalian species to human,
-  description of natural heterogeneity e.g. transmural variation of ion channel properties [ Antzelevitch, 2001 ],
-  description of arrhythmogenic substrates based on genetic defects [see e.g. Clancy and Rudy, 2002 ]
-  description of acquired arrhythmogenic substrates [see e.g. Priebe and Beuckelmann, 1998 ],
-  description of pathological changes in heterogeneity [see. e.g. Antzelevitch, 2001 ],
-  description of arrhythmogenic areas in or around ischemic or necrotic tissue [see e.g. Faber and Rudy, 2000 ].

Using advanced cell models the origin of abnormal automaticity Early After Depolarizations EAD, Delayed After Depolarizations DAD and dispersion of Action Potential Duration APD can be understood and influenced. More often the hypothesis about the origin of an arrhythmia is not a single arrhythmogenic zone but a problem in the interplay of cells ("mistuned orchestra"). This will be described in the next chapter.

4.  Coupling of Cells, Spread of Depolarization

Due to extremely large computation times for full bidomain 3D problems very often some of the simplifications mentioned in chapter 2 are used. Either e.g. a 2D-problem (atria) is modeled with the latest cell models [see e.g. Virag et al., 1998 ] or a 3D problem is simulated with a FitzHugh-Nagumo like model [see e.g. Berenfeld and Jalife, 1998; Rogers and McCulloch, 1994 ] or an eikonal wavefront equation [ Hunter and Pullan, 2002 ]. Either a monodomain approach is used for large areas or a full bidomain approach is tried out in small patches [see e.g. Seemann et al., 2001 ]. All this work is very important to learn about fundamental structures of arrhythmogenic processes (e.g. reentry, spiral waves), tendencies and ways to improve the method [ Winslow et al., 2001 ]. But care must be taken before results can be generalized. Important results so far are:

-  new insights to atrial fibrillation [see e.g. Nattel, 2002 ],
-  new views on ventricular tachycardia [see e.g. Panfilov, 1999 ],
-  new pictures on the role of gap junctions for reentry mechanisms [ Shaw and Rudy, 1997 ],
-  new aspects of heterogeneity on reentry arrhythmias [ Antzelevitch, 2001 ],
-  improved understanding of changes in wave propagation due to an obstacle or an isthmus [ Cabo et al., 1994 ]
-  new results on spiral wave formation, further development, and phase singularities [see e.g. Bray et al., 2001 ].

5.  External Stimulation

Many computer simulations start with the injection of a current into the intracellular space. Electrodes of pace makers and defibrillators on the other hand apply a current to the extracellular space. For a better understanding of this type of stimulation a detailed analysis is important [see e.g. Roth, 1994 ]. Two "classes" can be found in the literature. One class is describing all phenomena using the pure bidomain model [ Sepulveda and Wikswo, 1994; Wikswo et al., 1995 ]. This is perfect in case of highly conducting gap junctions. The other class is taking into account the cellular structure, leading to a hyperpolarization at the left end of the cells and a depolarization on the right side of the cells [ Keener and Panfilov, 1996; Krassowska et al., 1990; Trayanova, 1996 ]. Today it seems as if nearly all phenomena can be described based on the pure bidomain model although doubts have been raised again lately [ Sperelakis and McConnell, 2002 ].

Simulations of external stimulation that follow explicitly the time course of an ensemble of (fibrillating) cells will lead to a better understanding of the benefits of biphasic pulses and maybe to even better pulse forms for pace makers or defibrillators.

6.  Validation, Measurements

Extremely important work is going on in the field of validation. Fascinating results are presented by [ Muzikant et al., 2002 ] showing experiment and computer simulation going hand in hand. New measurement techniques allow for a fine adjustment of the virtual heart to better resemble the real heart:

-  electrode matrix to measure extracellular potentials in animal experiments with up to 500 channels [see e.g. Brooks and MacLeod, 1997 ],
-  electrode matrix to measure transmural extracellular potentials in animal experiments [see e.g. Ni et al., 1999 ],
-  electrode matrix to measure both extracellular and intracellular potentials in tissue probes [ Spach and Heidlage, 1994 ],
-  fluorescence optical techniques to determine the transmembrane voltage distribution "non-invasively" [see e.g. Lin and Wikswo, 1999; Nikolski and Efimov, 2001; Zhou et al., 1995 ],
-  Basket- and Balloon-electrodes that allow for multi-channel intracardial recordings both in animals and in patients [see e.g. Schilling et al., 2000 ].
-  Diffusion weighted MRI that can determine the fiber orientation in the heart of an animal. [ Muzikant et al., 2002 ].

Especially the measurements with patients are extremely important but unfortunately also extremely laborious. An individual heart and thorax model has to be created from MRI images, and the precise absolute location of the electrodes inside the body has to be determined.

7.  Conclusion

Substantial improvements have been achieved in the field of modeling leading to better understanding of arrhythmias and other pathologies. The gap between the virtual heart and the real heart becomes smaller and smaller. The benefit for the patient is not obvious yet. Computer models are already quite good in simulating the physiological case, but they lack behind in predicting the pathological case.

Acknowledgements

Substantial part of this work is based on research of Dr. Frank Sachse, who is now at Nora Eccles Harrison Cardiovascular Research and Training Institute.

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